Grothendieck Polynomial Formulas by Frank Sottile Berkeley Combinatorics Seminar 26 January 2004 Grothendieck polynomials were introduced by Lascoux and Schutzenberger as polynomial representatives of classes of Schubert structure sheaves in the Grothendieck ring of a flag variety. They are of considerable interest combinatorially as they are a natural generalization of Schubert polynomials. In this talk, I will discuss some formulas involving Grothendieck polynomials. These include (1) a Pieri-type formula for the multiplication of an arbitrary Grothendieck polynomial by one representing the structure sheaf of a special Schubert variety, and (2) a formula for expressing a Grothendieck polynomial as a sum of Grothendieck polynomials evaluated at subsets of the variables, where the coefficients are Schubert structure constants. This second formula involves permutation patterns and leads to a new construction of Grothendieck polynomials in terms of chains in the Bruhat order (easily seen to be equivalent to the standard formula in terms of rc-graphs.) It may be used to show that every Buch-Fulton quiver coefficient is, in a natural way, equal to a Schubert structure constant.